Axial vector and pseudo-vector seem to mean the same thing, and are dual to bivectors.

A bivector is a "directed area". (similar to how a vector is a "directed length") In three dimensions, a directed area can be represented by its normal vector (i.e. it's "axis"): that's where pseudovectors come from.

Similarly, in three dimensions, a "directed volume" can be represented by a number: that's where pseudoscalars come from.

Almost: [itex]A \times B[/itex] is the (pseudo)vector that is dual to the bivector [itex]A \wedge B[/itex].

[itex]A \times B[/itex] is merely the properly oriented vector that is perpendicular to the oriented parallelogram with sides A and B. Roughly speaking, [itex]A \wedge B[/itex] is that oriented parallelogram.